Derivations in the Banach ideals of $\tau$-compact operators

Abstract

Let $\mathcal{M}$ be a von Neumann algebra equipped with a faithful normal semi-finite trace $\tau$ and let $S_0(\tau)$ be the algebra of all $\tau$-compact operators affiliated with $\mathcal{M}$. Let $E(\tau)\subseteq S_0(\tau)$ be a symmetric operator space (on $\mathcal{M}$) and let $\mathcal{E}$ be a symmetrically-normed Banach ideal of $\tau$-compact operators in $\mathcal{M}$. We study (i) derivations $\delta$ on $\mathcal{M}$ with the range in $E(\tau)$ and (ii) derivations on the Banach algebra $\mathcal{E}$. In the first case our main results assert that such derivations are continuous (with respect to the norm topologies) and also inner (under some mild assumptions on $E(\tau)$). In the second case we show that any such derivation is necessarily inner when $\mathcal{M}$ is a type $I$ factor. As an interesting application of our results for the case (i) we deduce that any derivation from $\mathcal{M}$ into an $L_p$-space, $L_p(\mathcal{M},\tau)$, ($1<p<\infty$) associated with $\mathcal{M}$ is inner.